Calculus Examples

$y = \frac{x^{7}}{7} \cdot \ln (x) - \frac{x^{7}}{49}$

By the Sum Rule, the derivative of $\frac{x^{7}}{7} \cdot \ln (x) - \frac{x^{7}}{49}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{7}}{7} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{7}}{49}]$ .

$\frac{d}{d x} [\frac{x^{7}}{7} \cdot \ln (x)] + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Evaluate $\frac{d}{d x} [\frac{x^{7}}{7} \cdot \ln (x)]$ .

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Combine $\frac{x^{7}}{7}$ and $\ln (x)$ .

$\frac{d}{d x} [\frac{x^{7} \ln (x)}{7}] + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Since $\frac{1}{7}$ is constant with respect to $x$ , the derivative of $\frac{x^{7} \ln (x)}{7}$ with respect to $x$ is $\frac{1}{7} \frac{d}{d x} [x^{7} \ln (x)]$ .

$\frac{1}{7} \frac{d}{d x} [x^{7} \ln (x)] + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{7}$ and $g (x) = \ln (x)$ .

$\frac{1}{7} (x^{7} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{7}]) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{7} (x^{7} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{7}]) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$\frac{1}{7} (x^{7} \frac{1}{x} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Combine $x^{7}$ and $\frac{1}{x}$ .

$\frac{1}{7} (\frac{x^{7}}{x} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Cancel the common factor of $x^{7}$ and $x$ .

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Factor $x$ out of $x^{7}$ .

$\frac{1}{7} (\frac{x \cdot x^{6}}{x} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Cancel the common factors.

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Raise $x$ to the power of $1$ .

$\frac{1}{7} (\frac{x \cdot x^{6}}{x^{1}} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Factor $x$ out of $x^{1}$ .

$\frac{1}{7} (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Cancel the common factor.

$\frac{1}{7} (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Rewrite the expression.

$\frac{1}{7} (\frac{x^{6}}{1} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Divide $x^{6}$ by $1$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{d}{d x} [- \frac{x^{7}}{49}]$

Evaluate $\frac{d}{d x} [- \frac{x^{7}}{49}]$ .

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Since $- \frac{1}{49}$ is constant with respect to $x$ , the derivative of $- \frac{x^{7}}{49}$ with respect to $x$ is $- \frac{1}{49} \frac{d}{d x} [x^{7}]$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) - \frac{1}{49} \frac{d}{d x} [x^{7}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) - \frac{1}{49} (7 x^{6})$

Multiply $7$ by $- 1$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) - 7 (\frac{1}{49}) x^{6}$

Combine $- 7$ and $\frac{1}{49}$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{- 7}{49} x^{6}$

Combine $\frac{- 7}{49}$ and $x^{6}$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{- 7 x^{6}}{49}$

Cancel the common factor of $- 7$ and $49$ .

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Factor $7$ out of $- 7 x^{6}$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{7 (- x^{6})}{49}$

Cancel the common factors.

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Factor $7$ out of $49$ .

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{7 (- x^{6})}{7 (7)}$

Cancel the common factor.

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{7 (- x^{6})}{7 \cdot 7}$

Rewrite the expression.

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) + \frac{- x^{6}}{7}$

Move the negative in front of the fraction.

$\frac{1}{7} (x^{6} + \ln (x) (7 x^{6})) - \frac{x^{6}}{7}$

Simplify.

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Apply the distributive property.

$\frac{1}{7} x^{6} + \frac{1}{7} (\ln (x) (7 x^{6})) - \frac{x^{6}}{7}$

Combine terms.

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Combine $\ln (x)$ and $\frac{1}{7}$ .

$\frac{1}{7} x^{6} + \frac{\ln (x)}{7} (7 x^{6}) - \frac{x^{6}}{7}$

Combine $7$ and $\frac{\ln (x)}{7}$ .

$\frac{1}{7} x^{6} + \frac{7 \ln (x)}{7} x^{6} - \frac{x^{6}}{7}$

Combine $\frac{7 \ln (x)}{7}$ and $x^{6}$ .

$\frac{1}{7} x^{6} + \frac{7 \ln (x) x^{6}}{7} - \frac{x^{6}}{7}$

Cancel the common factor of $7$ .

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Cancel the common factor.

$\frac{1}{7} x^{6} + \frac{7 \ln (x) x^{6}}{7} - \frac{x^{6}}{7}$

Divide $\ln (x) x^{6}$ by $1$ .

$\frac{1}{7} x^{6} + \ln (x) x^{6} - \frac{x^{6}}{7}$

To write $\frac{1}{7} x^{6}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\ln (x) x^{6} + \frac{1}{7} x^{6} \cdot \frac{7}{7} - \frac{x^{6}}{7}$

Combine $\frac{1}{7} x^{6}$ and $\frac{7}{7}$ .

$\ln (x) x^{6} + \frac{\frac{1}{7} x^{6} \cdot 7}{7} - \frac{x^{6}}{7}$

Combine the numerators over the common denominator.

$\ln (x) x^{6} + \frac{\frac{1}{7} x^{6} \cdot 7 - x^{6}}{7}$

Combine $\frac{1}{7}$ and $x^{6}$ .

$\ln (x) x^{6} + \frac{\frac{x^{6}}{7} \cdot 7 - x^{6}}{7}$

Combine $\frac{x^{6}}{7}$ and $7$ .

$\ln (x) x^{6} + \frac{\frac{x^{6} \cdot 7}{7} - x^{6}}{7}$

Move $7$ to the left of $x^{6}$ .

$\ln (x) x^{6} + \frac{\frac{7 \cdot x^{6}}{7} - x^{6}}{7}$

Cancel the common factor of $7$ .

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Cancel the common factor.

$\ln (x) x^{6} + \frac{\frac{7 x^{6}}{7} - x^{6}}{7}$

Divide $x^{6}$ by $1$ .

$\ln (x) x^{6} + \frac{x^{6} - x^{6}}{7}$

Subtract $x^{6}$ from $x^{6}$ .

$\ln (x) x^{6} + \frac{0}{7}$

Cancel the common factor of $0$ and $7$ .

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Factor $7$ out of $0$ .

$\ln (x) x^{6} + \frac{7 (0)}{7}$

Cancel the common factors.

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Factor $7$ out of $7$ .

$\ln (x) x^{6} + \frac{7 \cdot 0}{7 \cdot 1}$

Cancel the common factor.

$\ln (x) x^{6} + \frac{7 \cdot 0}{7 \cdot 1}$

Rewrite the expression.

$\ln (x) x^{6} + \frac{0}{1}$

Divide $0$ by $1$ .

$\ln (x) x^{6} + 0$

Add $\ln (x) x^{6}$ and $0$ .

$\ln (x) x^{6}$

Reorder the factors of $\ln (x) x^{6}$ .

$x^{6} \ln (x)$